Problem:
Ten identical crates each have dimensions . The first crate is placed flat on the floor. Each of the remaining nine crates is placed, in turn, flat on top of the previous crate, and the orientation of each crate is chosen at random. Let be the probability that the stack of crates is exactly tall, where and are relatively prime positive integers. Find .
Solution:
Because each crate has three different possible heights, there are equally likely ways to stack the ten crates. Suppose a stack of crates is tall. Let , and be the number of crates with heights , and , respectively. Then and . This system is equivalent to the system and , or , . Because and , with each value of yielding a different solution :
For each ordered triple there are equally likely ways to stack the crates. Thus the number of ways to stack the crates is . The desired probability is , so the required value of is .
Note: Another way to find the ordered triples which solve the two given equations is to notice that a stack of crates must be at least tall. Each crate adds additional foot to the height, and each crate adds additional feet. The stack requires more feet than the minimum of feet, so .
The problems on this page are the property of the MAA's American Mathematics Competitions